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Monoidal computer is a categorical model of intensional computation, wheremany different programs correspond to the same input-output behavior. Theupshot of yet another model of computation is that a categorical formalismshould provide a much needed high level language for theory of computation,flexible enough to allow abstracting away the low level implementation detailswhen they are irrelevant, or taking them into account when they are genuinelyneeded. A salient feature of the approach through monoidal categories is theformal graphical language of string diagrams, which supports visual reasoningabout programs and computations.  In the present paper, we provide a coalgebraic characterization of monoidalcomputer. It turns out that the availability of interpreters and specializers,that make a monoidal category into a monoidal computer, is equivalent with theexistence of a *universal state space*, that carries a weakly final statemachine for any pair of input and output types. Being able to program statemachines in monoidal computers allows us to represent Turing machines, tocapture their execution, count their steps, as well as, e.g., the memory cellsthat they use. The coalgebraic view of monoidal computer thus provides aconvenient diagrammatic language for studying computability and complexity.

Monoidal computer III: A coalgebraic view of computability and complexity